Semi-fractality Discovered in Quantum Particles on Chiral Cayley Trees

Researchers have discovered a new type of wave-function behavior, termed semi-fractality, in quantum particles hopping on a specific lattice structure known as a chiral Cayley tree. This model, which lacks on-site disorder but features nearest-neighbor hopping amplitudes drawn from a singular distribution, reveals that the particle's eigenstates occupy an extensive fraction of the system. However, their higher moments exhibit characteristics typical of a multifractal state, implying an unusual wave-function statistics. The study utilized population dynamics to solve the cavity equations for the propagator, enabling an analysis of the local density of states distribution. This distribution was found to develop broad power-law tails, indicating the semi-fractal nature of the wave functions. The chiral symmetry of the model, inherent to its bipartite nature, significantly influences the statistics of eigenstates at the center of the energy spectrum, a crucial aspect for understanding this behavior. A key finding is that the symmetry properties of the local density of states distribution are not solely fixed by the symmetry class but vary continuously with the exponent controlling the power-law hopping distribution. As this exponent is changed, the system transitions from a semi-fractal regime to a localized one. At the transition point, the wave functions adopt an extreme intermediate form, termed semi-localized, which is simultaneously extended in its support but localized according to its higher moments. This discovery opens new avenues for understanding quantum localization and delocalization in complex systems.

Benchmarking Methods for Bubble Wall Velocities in Cosmological Phase Transitions

Researchers have compared two key computational approaches, the "fluid Ansatz" and the "WallGo" code, to determine the velocity of bubble walls in cosmological phase transitions. These transitions are crucial for understanding the early universe, and the bubble velocity ($v_w$) is a fundamental parameter for predicting gravitational wave signals. The study reveals that both methods agree closely in the regime of reasonably mild phase transitions, with a strength parameter $\alpha \lesssim 0.01$. The agreement is particularly good when only top-quark annihilation is considered. However, a noticeable discrepancy appears once scattering processes are included. The work also investigates the limitations of linearizing the Boltzmann equation when applying the fluid Ansatz to stronger phase transitions. It is observed that non-linear contributions induce significant shifts in the predicted terminal velocity as $\alpha \to 1$, even though the non-linear contribution to the wall pressure remains quantitatively small compared to the equilibrium and linearized non-equilibrium parts. These findings have important implications for the interpretation of future gravitational wave observations. Strong phase transitions are precisely the primary targets for future gravitational wave observatories. Therefore, the study emphasizes the need for higher precision computations of $v_w$ in the semi-classical approach and suggests that a treatment beyond the WKB approximation may be needed to adequately understand these extreme early universe events.